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During the epoch of ninety years from 1730 to 1820 the French and Swiss cultivated mathematics with most brilliant success. No previous period had shown such an array of illustrious names. At this time Switzerland had her Euler; France, her Lagrange, Laplace, Legendre, and Monge. The mediocrity of French mathematics which marked the time of Louis XIV. was now followed by one of the very brightest periods of all history. England and Germany, on the other hand, which during the unproductive period in France had their Newton and Leibniz, could now boast of no great mathematician. France now waved the mathematical sceptre. Mathematical studies among the English and German people had sunk to the lowest ebb. Among them the direction of original research was ill-chosen. The former adhered with excessive partiality to ancient geometrical methods; the latter produced the combinatorial school, which brought forth nothing of value.

The labours of Euler, Lagrange, and Laplace lay in higher analysis, and this they developed to a wonderful degree. By them analysis came to be completely severed from geometry. During the preceding period the effort of mathematicians not only in England, but, to some extent, even on the continent, had been directed toward the solution of problems clothed in geometric garb, and the results of calculation were usually reduced to geometric form. A change now took place. Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected